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I'm not a course instructor, just a TA of the first quarter calculus course who lead discussion sections and grade exams.

When grading the midterm, I found large number of students showed some understanding of the concepts/problem solving strategies, but writing them using poor notations.

For example,

Question: Find the horizontal asymptotes for the function $h(x)=\frac{x-1}{x^2-6x+8}$

Some students' answers are similar to

Answer: $$\lim_{x\to \infty}\frac{x-1}{x^2-6x+8}=\lim_{x\to \infty}\frac{x-1}{(x-2)(x-4)}=\frac{\infty}{\infty\cdot\infty}=\frac{1}{\infty}=0$$ so H.A. at 0.

Answers like this usually

(a). take lots of time to grade and hard to decide how many points (out of 10, for example) should be given. (Imagine yourself trying to grade consistently and seeing 10 different variations of answers like that in a 300 people class)

(b). result in students submitting regrade requests asking why 6 points instead of 7 points are given, therefore extra workload for TA.

So my question is, in the discussion section I'm leading, what teaching strategy can I use to reduce the amount of answers like that in the future exams I'm grading?

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    $\begingroup$ Have you talked about this issue with the course instructor? It might be wise to confer with them before you implement any novel teaching strategies (e.g., strategies to streamline notation) lest the students end up with inconsistent expectations around different routines. $\endgroup$ – Benjamin Dickman Oct 29 '15 at 9:10
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    $\begingroup$ If the students have had explicit instruction and practice in using some more rigorous technique for doing this, then I don't see why this should be worth any partial credit. By the time I asked my calc 1 students to do a problem like this, they would have seen and practiced multiple methods of doing it (picking out dominant terms, dividing out the highest power on top and bottom, l'Hospital's rule). Why give any credit for an answer that simply shows they failed to learn the techniques they were supposed to learn? Partial credit is for work that is partially right, not for work that's wrong. $\endgroup$ – Ben Crowell Oct 29 '15 at 14:44
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    $\begingroup$ For me in a scale of 10, an answer like that gets consistenly 0. $\endgroup$ – Massimo Ortolano Oct 30 '15 at 17:03
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    $\begingroup$ @MassimoOrtolano I'm not sure where you teach and what instructors you teach with, but my experience from a high quality (one of the top 10 universities in math in the US in multiple years) US university is that it is not feasible to grade that problem as 0. Believe me I'd have loved to be able to do that since I come from a background where if you put that down not only would you get 0 points on the question but would be told you have to retake the test since you obviously have no clue what you are doing. Unfortunately you have to accommodate the local culture. $\endgroup$ – DRF Dec 10 '15 at 14:43
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    $\begingroup$ @DRF I teach in Italy. I'm the instructor (in my country TAs don't grade exams so frequently), and along 15 years of teaching I gave many zeros (though in an another field, not mathematics). I'm not happy about giving a zero, but if the exercise is not salvageable in any way, that's it. However, in Italy you can retake an exam until you pass it (we have 3-4 exam sessions along the year), and it's not uncommon to have students who retake an exam at least 2-3 times (sometimes even 10 or more). $\endgroup$ – Massimo Ortolano Dec 10 '15 at 15:31
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Let me echo Benjamin's comment that any proactive step that you take should be done with the instructor's permission.

At a practical level, I think there are ways to address issues (a) and (b). For (a), make a rubric (either in advance or a running one as you go) which lays out the criteria for awarding points. This allows you to be consistent with how you distribute points and, since the assignment of points itself is subjective in nature, a reasonable rubric enforced as consistently as possible usually results in good grading. Your instructor can give you a sense of whether your overall grading standards are too harsh or too lenient. For (b), if you have followed (a) and have the support of your instructor, I think it would be a reasonable policy not to consider changes to homework grades unless a substantial error was made (e.g. you misread a student's response and marked a correct method as being incorrect).

But regardless of grading issues, we would certainly like calculus students to produce better responses than the one you used as an example. In my view, one needs to take a perspective similar to triage in medicine. Often, students have many deficiencies relative to the level of performance we would like to see from them, so prioritizing the most important issues to address is essential to good instruction. Since examples are often easier to discuss than general principles, I'll stick with your specific example.

I see one glaring issue that is very concerning, and that is the use of the step $\frac{\infty}{\infty \cdot \infty} = \frac{1}{\infty}$. Not only is it an invalid thought process, it is likely to produce incorrect conclusions if the student attempts to extend it to other contexts. Would $\lim\limits_{x \rightarrow \infty} \frac{(x - 1)(x + 4)}{x^3}$ equal $\frac{\infty \cdot \infty}{\infty}$ or $\frac{\infty \cdot \infty}{\infty^3}$? What would happen if the denominator is $\sqrt[4]{x}^3$ instead of $x^3$? It's not clear that the student would use the right "power" of $\infty$ to be able to compute the limit. (If they are able to, that's a positive sign -- at least they have some sense of what is going on, even if their articulation of the reasoning is very flawed.) In comparison, if a student writes $\lim\limits_{x \rightarrow \infty} \frac{1}{x} = \frac{1}{\infty} = 0$, I'm not thrilled with the inappropriate use of $\infty$ (according to the standard definitions), but I consider it mostly harmless. There is reasonable sense that can be made to $\frac{1}{\infty} = 0$ that is much more palatable than nonsense such as $\frac{\infty}{\infty \cdot \infty} = 0$.

The other issue that I would want to address is that "H.A. at 0" is not very precise and I would prefer a student write that "the line $y = 0$ is a horizontal asymptote" ("H.A." in place of "horizontal asymptote" is fine), but some instructors may prefer the less formal means of expression. After all, someone could be pedantic with my approach and say that $y = 0$ is not actually a line, so we should write "the line which is the locus of all points where $y = 0$" or some such and now we've simply confused the students entirely. In practice, I would accept an answer such as "H.A. at 0" from students, but expect myself and my TAs to be more precise and refer to the line $y = 0$ as the horizontal asymptote.

Once you've identified the issues you want to address, here are some concrete ways you can obtain improvements:

Be sure you are very careful in your own explanations of solutions. I've had TAs who wanted to mark very harshly for sloppy writing on homework ... and then I observed them engage in the same sloppiness on the board during their recitation periods. When I brought this issue up, they argued, "Well of course I have to be sloppy in class because I am under time pressure, but students doing homework have the time to write better." The flaws in this reasoning are many, but in brief (1) it is generally better to write explanations clearly on the board and cover less problems than to sloppily solve many problems; (2) students cannot possibly learn to write well if they do not see a consistent high standard [and certainly not if they see sloppiness is acceptable for an instructor or TA]; (3) students do not have the time or do not choose to use their time to write polished solutions to every calculus problem they are assigned -- and it's not realistic to expect that [so if good writing has not become a learned habit, don't expect it to happen].

So you start by providing a good example yourself. Then you perform triage and identify the worst offenses that you will mark off on homework. The cancellation of factors of $\infty$ in your example would certainly qualify. You should take time to address common mistakes. I recommend a brief amount of time addressing such issues in class (a list of things not to do and 1-2 sentence explanations of why for each item on the list), with an offer to explain more thoroughly the rationale in office hours. Students will respond to the incentive of losing points, and they are generally accepting if you can show that there is a reasonable rationale behind it (even if they don't fully understand or accept that rationale). Just don't appeal to authority or superiority. If you cannot give an explanation that a reasonably strong student can understand, you are probably setting an unrealistic expectation.

As a TA, it should be added, that you should generally rely on your instructor's opinion to perform your triage assessment. Certainly it is inappropriate to simply decide your own priorities without any consultation of the instructor. The instructor often cannot micromanage every single prioritization decision you make, but a good conversation with your instructor can leave you with a good understanding of how to prioritize (and a good instructor will understand that at the level of small details, you two may not come to the exact same conclusions and that is okay as long as you are in harmony overall).

Ideally, by the time exams come around, students have had good instruction to see what an appropriate solution looks like for the types of problems they encounter, and the homework has identified the most egregious errors in writing up solutions. Then the markings for exams should generally follow the same standards. (In practice, I prefer to be have a somewhat more harsh grading standard for homework than exams. Harsh feedback on homework is more likely to get their attention and lead to improvement before the next exam. And that improvement will hopefully lead to increased retention of conceptual understanding for use later in the course and beyond.)

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    $\begingroup$ @fmlin. Thanks for the clarification. I understand the need for electronic grading of homework, especially as class sizes increase in many universities, and this policy is presumably out of your control. You can grade work and indicate improper reasoning on the weekly quizzes. Realistically, few students will be able to write up proper solutions consistently without a good deal of feedback on their attempts. Another option is to provide solutions for selected hw problems, with attention given to dispelling common misconceptions and sources of error. $\endgroup$ – Michael Joyce Oct 29 '15 at 18:04
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    $\begingroup$ I do not agree that the manipulation of infinities should be considered more acceptable even if the student can use the right "powers" of it. The reason is that asymptotic growth is not the same as equality, so using "$\lim$" in the left-hand expression but $\infty$ in the right-hand expression is a clear sign of an intrinsically flawed understanding, since the only way to make sense of $\infty$ in an equality is to use the extended real line, but then $\infty \div \infty$ is not allowed. $\endgroup$ – user21820 Oct 30 '15 at 4:16
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    $\begingroup$ @MichaelJoyce: "calculus is one of the worst subjects to use for first introducing students to rigor and careful logical thinking" - In Germany (and I guess most of Europe), students learn first about calculus (and a whole bunch of other things) in high school where nothing they do is very rigorous anyway. However, when they go to university - and they do maths or anything scientific - they will usually relearn calculus (or rather, analysis) from first principles with an emphasis on rigour in the first semester. So I'm not sure about your assertion - at least not in university. $\endgroup$ – Fryie Nov 11 '15 at 19:57
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    $\begingroup$ @Fryie: That's a sensible system. In the US, students have a range of experience in high school mathematics, from no calculus to a large base of calculus knowledge (almost never rigorous). At my university, we have all of those backgrounds in one course. In practice, that makes it impossible to teach the desired content in a fully rigorous manner in a way that is accessible to students with no or limited background in calculus. Or at least, that is my experience. I certainly would prefer your situation to the one I currently deal with. $\endgroup$ – Michael Joyce Nov 11 '15 at 20:15
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    $\begingroup$ Right, I see where you're coming from. Traditionally, here, there are different courses for different majors. For example, economists have their own mathematics courses that, afaik, don't put much emphasis on rigour. For people who only ever use applications that might make a lot of sense. As a maths major, I would probably feel I'd be wasting my time learning about applications I don't care about instead of fundamentals, though. $\endgroup$ – Fryie Nov 12 '15 at 12:11
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Michael Joyce's answer above is excellent. Let me sketch out a few extra thoughts. Assume in principle that you're the instructor, or else that you can discuss and come to agreement with the instructor on the following:

The points assessed on each item should be scaled in advance to the length of the item, such that grading rubric is either self-evident or easy to specify, and the grader wastes no time in debate on what points to take off where. In the example problem, I wouldn't assign it more than 5 points, since it only takes about 5 steps of work to generate the answer; since the middle 3 steps are all clearly fallacious, I'd probably take off 3 points in this example.

If required to assess it at 10 points, then I'd grade in 5 2-point increments each, and take off 6 points for the example work; no need to sweat over finer granularity than that. (Personally, I would probably set up this problem at only 3 points on my own tests.) Furthermore, distribute a complete answer sheet with grading rubric (just note how many points at each step), so that (a) students can see the complete, correct answer, and (b) ascertain that the grading was done in a reasonably objective manner -- hopefully heading off complaints or requests for re-grading.

As Michael indicates, definitely model the proper writing in class, and question/ask for directions on proper writing at each step from the class. I do think that grading the details is the only way to get students to attend to proper care in their writing and reasoning.

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In my own lectures, there are a couple things that I have found to be useful for addressing notation issues in math (especially calculus). Personally, I do a minimum of "correcting" while grading exams outside of what will be necessary to reconstruct why points were taken off. For example, I might just circle the $\infty$ stretch and put "-3" next to it. The students will presumably get an answer key and correcting such mistakes really requires an interaction rather than a paragraph written on an exam.

  1. Make it clear from the start that notation is important, just as spelling and grammar are important for writing. The time for correcting notation is not on the exam proper, but rather assessments leading up to the exam and class discussions following the exam. There does, however, need to be an acknowledgement of some leeway. For example, the posed question on the exam is grammatically incorrect---there's no period at the end of the sentence. How many points should the exam writer lose?
  2. Be thoroughly meticulous on the board. It would be hard to learn a foreign language if the instructor mis-spelled many words. This means that every instance of $\lim$ is accompanied by bounds, bounds don't vanish from integrals, etc. Every. Single. Time. I've even taken this to the extent of "$2+2=4.$", as there needs to be a period at the end of mathematical sentences. This is not to say that the instructor is to be blamed, but rather that a high level of rigor allows you to be beyond reproach when taking points off for notation.
  3. Emphasize why notation matters by analogy with more familiar topics. Perhaps I've noticed some of my students writing $\lim_{x \to \infty} =0$ or $\frac{\mathrm{d}}{\mathrm{d}x} = 2x$. I might start a class with writing $+ = 3$ on the board and asking them to solve. Once they've agreed that such a string of symbols is ridiculous, circle back to calculus and emphasize that the calculus-y symbols are operators.
  4. The particular answer you have given is problematic as we run into the issue of the correct answer with incorrect reasoning. Taking points off (as you should) will invariably lead to the complaint "But I got the right answer!" If possible, write questions where $\frac{\infty}{\infty^2}$ type notation will lead to the wrong answer.
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As far as I see, most of you disagree, but I do not find $\frac{\infty}{\infty \cdot \infty}$ wrong. I could write it myself. It leads to the correct answer, and not by accident. If the student can not use this method in other cases, he will lose points when the exam questions include them. Here, in the worst case, I would understand noting that this notation is not very precise and requires quotation marks (or some talking about nonstandard analysis and probably using other sign than $\infty$), but myself I would just write "a bit risky trick" without subtracting points. Would you give the student full points if he did not write this two steps with infinity? Why punish him for showing his reasoning?

On the other hand, as the final answer I would expect at least something like "two-sided horizontal asymptotics y=0" or "2-s. H.A. y=0". The answer given does not show that the student knows that there can be two horizontal asimptotes (even $\rightarrow \pm \infty$ in the first limit could be enough) and that he knows what horizontal asymptotics is - he could just write the number that he gets from the formula. "At" is a good sign, so I would give 8.5/10: -1 for no signs of seeing the possibility two asymptotes and -0.5 for giving an answer that looks rather like giving random number.

I find understending (which he does not show clear enough) and correct answers (which he gives) more important than not writing something that is against standard rules, but makes sense.

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    $\begingroup$ I downvoted because $\displaystyle \frac{\infty}{\infty \cdot \infty}$ is wrong. $\endgroup$ – Mark Fantini Nov 3 '15 at 11:30
  • $\begingroup$ @MarkFantini: I could expect such criticism. Note however that if we redefine $\infty$ as a sign of a class of infinite hyperreal numbers that differ by finte values and $=$ as "difference is infinitesimal", everything is correct. This is not worse than more accepted notations like $5 x^2+2=O(x^2)$ or $\int_0^x x dx = \frac{1}{2} x^2$. $\endgroup$ – BartekChom Nov 3 '15 at 11:56
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    $\begingroup$ I'd say that in mathematics, everything is undefined unless introduced explicitly. I don't assume hyperreals were mentioned in class, nor is it probable that the student introduced them in the answer. I think you want students to be able to do mathematics rigourously. If they use some non-standard method for it, fine, but just using "tricks" without understanding them / having established that they work, shouldn't be given credit. $\endgroup$ – Fryie Nov 3 '15 at 16:04
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    $\begingroup$ @BartekChom: Writing $\infty$ for every randomly chosen hyperreal is exactly one of those things that NSA tells you that you cannot do. NSA does allow you to manipulate infinite quantities according to the elementary axioms of the real number system, but it does not say that all infinite quantities are equal in size and therefore representable by the same symbol. As a counterexample, consider $\lim_{x\rightarrow\infty}(e^x/(x\sqrt{x}))$. $\endgroup$ – Ben Crowell Nov 3 '15 at 16:11
  • $\begingroup$ @Fryie: OK. I see that most mathematicians find rigour very important and understand that you do not want to give credits for anything not explicitely defined. Hovever, this solution suggests that the student has reinvented at least some properties of hyperreals - they are very intuitive, so it is not strange. $\endgroup$ – BartekChom Nov 4 '15 at 5:38

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